dx Problem 2) (45 Points) Consider the boundary value problem solved above: do ODE: :-10 = 0 on 0

Problem 2) (45 Points) Consider the boundary value problem solved above: a. b. c. ODE: = 0 on 0 < x < L with BC's: +(0) = o and t: (0) = o and (15 Points) Derive the first form of Rayleigh quotient directly from the statement of the eigenvalue problem by multiplying the ODE by the eigenfunction, integrating over the domain, and solving for A. (20 Points) Obtain the second form of the Rayleigh quotient by integrating the numerator twice by parts and using the boundary conditions. (10 Points) Using ONLY the Rayleigh quotient from b. and the boundary conditions, i. show that all the eigenvalues are non-negative, i.e., show that A 0, ii. show that a nontrivial solution for (x) can exist if A = O by setting the numerator equal to zero and determining limitations this has on $